src/HOL/GCD.thy
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(*  Title:      GCD.thy
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    Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad
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This file deals with the functions gcd and lcm, and properties of
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primes. Definitions and lemmas are proved uniformly for the natural
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numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chiaeb.
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*)
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header {* GCD *}
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theory GCD
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imports NatTransfer
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begin
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declare One_nat_def [simp del]
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subsection {* gcd *}
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class gcd = one +
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fixes
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  gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
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  lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation
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  coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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where
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  "coprime x y == (gcd x y = 1)"
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end
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class prime = one +
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  prime :: "'a \<Rightarrow> bool"
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(* definitions for the natural numbers *)
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instantiation nat :: gcd
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begin
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fun
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  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "gcd_nat x y =
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   (if y = 0 then x else gcd y (x mod y))"
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definition
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  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "lcm_nat x y = x * y div (gcd x y)"
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instance proof qed
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end
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instantiation nat :: prime
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begin
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definition
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  prime_nat :: "nat \<Rightarrow> bool"
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where
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  [code del]: "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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instance proof qed
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end
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(* definitions for the integers *)
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instantiation int :: gcd
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begin
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definition
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  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
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where
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  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
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definition
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  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
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where
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  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
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instance proof qed
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end
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instantiation int :: prime
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begin
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definition
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  prime_int :: "int \<Rightarrow> bool"
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where
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  [code del]: "prime_int p = prime (nat p)"
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instance proof qed
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end
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subsection {* Set up Transfer *}
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lemma transfer_nat_int_gcd:
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  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
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  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
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  "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
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  unfolding gcd_int_def lcm_int_def prime_int_def
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  by auto
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lemma transfer_nat_int_gcd_closures:
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare TransferMorphism_nat_int[transfer add return:
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    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
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lemma transfer_int_nat_gcd:
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  "gcd (int x) (int y) = int (gcd x y)"
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  "lcm (int x) (int y) = int (lcm x y)"
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  "prime (int x) = prime x"
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  by (unfold gcd_int_def lcm_int_def prime_int_def, auto)
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lemma transfer_int_nat_gcd_closures:
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare TransferMorphism_int_nat[transfer add return:
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    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
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subsection {* GCD *}
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(* was gcd_induct *)
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lemma nat_gcd_induct:
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  fixes m n :: nat
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  assumes "\<And>m. P m 0"
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    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
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  shows "P m n"
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  apply (rule gcd_nat.induct)
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  apply (case_tac "y = 0")
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  using assms apply simp_all
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done
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(* specific to int *)
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lemma int_gcd_neg1 [simp]: "gcd (-x::int) y = gcd x y"
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  by (simp add: gcd_int_def)
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lemma int_gcd_neg2 [simp]: "gcd (x::int) (-y) = gcd x y"
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  by (simp add: gcd_int_def)
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lemma int_gcd_abs: "gcd (x::int) y = gcd (abs x) (abs y)"
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  by (simp add: gcd_int_def)
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lemma int_gcd_cases:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
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  shows "P (gcd x y)"
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by (insert prems, auto simp add: int_gcd_neg1 int_gcd_neg2, arith)
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lemma int_gcd_ge_0 [simp]: "gcd (x::int) y >= 0"
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  by (simp add: gcd_int_def)
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lemma int_lcm_neg1: "lcm (-x::int) y = lcm x y"
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  by (simp add: lcm_int_def)
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lemma int_lcm_neg2: "lcm (x::int) (-y) = lcm x y"
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  by (simp add: lcm_int_def)
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lemma int_lcm_abs: "lcm (x::int) y = lcm (abs x) (abs y)"
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  by (simp add: lcm_int_def)
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lemma int_lcm_cases:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
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  shows "P (lcm x y)"
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by (insert prems, auto simp add: int_lcm_neg1 int_lcm_neg2, arith)
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lemma int_lcm_ge_0 [simp]: "lcm (x::int) y >= 0"
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  by (simp add: lcm_int_def)
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(* was gcd_0, etc. *)
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lemma nat_gcd_0 [simp]: "gcd (x::nat) 0 = x"
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  by simp
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(* was igcd_0, etc. *)
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lemma int_gcd_0 [simp]: "gcd (x::int) 0 = abs x"
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  by (unfold gcd_int_def, auto)
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lemma nat_gcd_0_left [simp]: "gcd 0 (x::nat) = x"
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  by simp
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lemma int_gcd_0_left [simp]: "gcd 0 (x::int) = abs x"
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  by (unfold gcd_int_def, auto)
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lemma nat_gcd_red: "gcd (x::nat) y = gcd y (x mod y)"
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  by (case_tac "y = 0", auto)
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   235
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(* weaker, but useful for the simplifier *)
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lemma nat_gcd_non_0: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
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  by simp
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lemma nat_gcd_1 [simp]: "gcd (m::nat) 1 = 1"
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  by simp
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lemma nat_gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
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  by (simp add: One_nat_def)
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lemma int_gcd_1 [simp]: "gcd (m::int) 1 = 1"
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  by (simp add: gcd_int_def)
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lemma nat_gcd_self [simp]: "gcd (x::nat) x = x"
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  by simp
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lemma int_gcd_def [simp]: "gcd (x::int) x = abs x"
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  by (subst int_gcd_abs, auto simp add: gcd_int_def)
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declare gcd_nat.simps [simp del]
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text {*
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  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
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  conjunctions don't seem provable separately.
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*}
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lemma nat_gcd_dvd1 [iff]: "(gcd (m::nat)) n dvd m"
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  and nat_gcd_dvd2 [iff]: "(gcd m n) dvd n"
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  apply (induct m n rule: nat_gcd_induct)
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  apply (simp_all add: nat_gcd_non_0)
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  apply (blast dest: dvd_mod_imp_dvd)
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done
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thm nat_gcd_dvd1 [transferred]
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lemma int_gcd_dvd1 [iff]: "gcd (x::int) y dvd x"
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  apply (subst int_gcd_abs)
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  apply (rule dvd_trans)
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  apply (rule nat_gcd_dvd1 [transferred])
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  apply auto
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done
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lemma int_gcd_dvd2 [iff]: "gcd (x::int) y dvd y"
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  apply (subst int_gcd_abs)
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  apply (rule dvd_trans)
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  apply (rule nat_gcd_dvd2 [transferred])
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  apply auto
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done
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lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
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by(metis nat_gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
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by(metis nat_gcd_dvd2 dvd_trans)
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lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
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by(metis int_gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
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by(metis int_gcd_dvd2 dvd_trans)
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lemma nat_gcd_le1 [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
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  by (rule dvd_imp_le, auto)
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lemma nat_gcd_le2 [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
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  by (rule dvd_imp_le, auto)
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lemma int_gcd_le1 [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
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  by (rule zdvd_imp_le, auto)
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lemma int_gcd_le2 [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
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  by (rule zdvd_imp_le, auto)
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   310
lemma nat_gcd_greatest: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   311
  by (induct m n rule: nat_gcd_induct) (simp_all add: nat_gcd_non_0 dvd_mod)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   312
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   313
lemma int_gcd_greatest:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   314
  assumes "(k::int) dvd m" and "k dvd n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   315
  shows "k dvd gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   316
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   317
  apply (subst int_gcd_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   318
  apply (subst abs_dvd_iff [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   319
  apply (rule nat_gcd_greatest [transferred])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   320
  using prems apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   321
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   322
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   323
lemma nat_gcd_greatest_iff [iff]: "(k dvd gcd (m::nat) n) =
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   324
    (k dvd m & k dvd n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   325
  by (blast intro!: nat_gcd_greatest intro: dvd_trans)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   326
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   327
lemma int_gcd_greatest_iff: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   328
  by (blast intro!: int_gcd_greatest intro: dvd_trans)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   329
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   330
lemma nat_gcd_zero [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   331
  by (simp only: dvd_0_left_iff [symmetric] nat_gcd_greatest_iff)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   332
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   333
lemma int_gcd_zero [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   334
  by (auto simp add: gcd_int_def)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   335
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   336
lemma nat_gcd_pos [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   337
  by (insert nat_gcd_zero [of m n], arith)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   338
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   339
lemma int_gcd_pos [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   340
  by (insert int_gcd_zero [of m n], insert int_gcd_ge_0 [of m n], arith)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   341
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   342
lemma nat_gcd_commute: "gcd (m::nat) n = gcd n m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   343
  by (rule dvd_anti_sym, auto)
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   344
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   345
lemma int_gcd_commute: "gcd (m::int) n = gcd n m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   346
  by (auto simp add: gcd_int_def nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   347
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   348
lemma nat_gcd_assoc: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   349
  apply (rule dvd_anti_sym)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   350
  apply (blast intro: dvd_trans)+
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   351
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   352
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   353
lemma int_gcd_assoc: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   354
  by (auto simp add: gcd_int_def nat_gcd_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   355
31766
f767c5b1702e new lemmas
nipkow
parents: 31734
diff changeset
   356
lemmas nat_gcd_left_commute =
f767c5b1702e new lemmas
nipkow
parents: 31734
diff changeset
   357
  mk_left_commute[of gcd, OF nat_gcd_assoc nat_gcd_commute]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   358
31766
f767c5b1702e new lemmas
nipkow
parents: 31734
diff changeset
   359
lemmas int_gcd_left_commute =
f767c5b1702e new lemmas
nipkow
parents: 31734
diff changeset
   360
  mk_left_commute[of gcd, OF int_gcd_assoc int_gcd_commute]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   361
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   362
lemmas nat_gcd_ac = nat_gcd_assoc nat_gcd_commute nat_gcd_left_commute
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   363
  -- {* gcd is an AC-operator *}
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   364
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   365
lemmas int_gcd_ac = int_gcd_assoc int_gcd_commute int_gcd_left_commute
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   366
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   367
lemma nat_gcd_1_left [simp]: "gcd (1::nat) m = 1"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   368
  by (subst nat_gcd_commute, simp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   369
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   370
lemma nat_gcd_Suc_0_left [simp]: "gcd (Suc 0) m = Suc 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   371
  by (subst nat_gcd_commute, simp add: One_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   372
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   373
lemma int_gcd_1_left [simp]: "gcd (1::int) m = 1"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   374
  by (subst int_gcd_commute, simp)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   375
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   376
lemma nat_gcd_unique: "(d::nat) dvd a \<and> d dvd b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   377
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   378
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   379
  apply (rule dvd_anti_sym)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   380
  apply (erule (1) nat_gcd_greatest)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   381
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   382
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   383
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   384
lemma int_gcd_unique: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   385
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   386
  apply (case_tac "d = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   387
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   388
  apply (rule iffI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   389
  apply (rule zdvd_anti_sym)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   390
  apply arith
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   391
  apply (subst int_gcd_pos)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   392
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   393
  apply (drule_tac x = "d + 1" in spec)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   394
  apply (frule zdvd_imp_le)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   395
  apply (auto intro: int_gcd_greatest)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   396
done
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 30042
diff changeset
   397
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   398
text {*
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   399
  \medskip Multiplication laws
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   400
*}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   401
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   402
lemma nat_gcd_mult_distrib: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   403
    -- {* \cite[page 27]{davenport92} *}
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   404
  apply (induct m n rule: nat_gcd_induct)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   405
  apply simp
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   406
  apply (case_tac "k = 0")
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   407
  apply (simp_all add: mod_geq nat_gcd_non_0 mod_mult_distrib2)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   408
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   409
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   410
lemma int_gcd_mult_distrib: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   411
  apply (subst (1 2) int_gcd_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   412
  apply (simp add: abs_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   413
  apply (rule nat_gcd_mult_distrib [transferred])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   414
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   415
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   416
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   417
lemma nat_gcd_mult [simp]: "gcd (k::nat) (k * n) = k"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   418
  by (rule nat_gcd_mult_distrib [of k 1 n, simplified, symmetric])
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   419
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   420
lemma int_gcd_mult [simp]: "gcd (k::int) (k * n) = abs k"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   421
  by (rule int_gcd_mult_distrib [of k 1 n, simplified, symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   422
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   423
lemma nat_coprime_dvd_mult: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   424
  apply (insert nat_gcd_mult_distrib [of m k n])
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   425
  apply simp
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   426
  apply (erule_tac t = m in ssubst)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   427
  apply simp
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   428
  done
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   429
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   430
lemma int_coprime_dvd_mult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   431
  assumes "coprime (k::int) n" and "k dvd m * n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   432
  shows "k dvd m"
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   433
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   434
  using prems
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   435
  apply (subst abs_dvd_iff [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   436
  apply (subst dvd_abs_iff [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   437
  apply (subst (asm) int_gcd_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   438
  apply (rule nat_coprime_dvd_mult [transferred])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   439
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   440
  apply (subst abs_mult [symmetric], auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   441
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   442
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   443
lemma nat_coprime_dvd_mult_iff: "coprime (k::nat) n \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   444
    (k dvd m * n) = (k dvd m)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   445
  by (auto intro: nat_coprime_dvd_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   446
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   447
lemma int_coprime_dvd_mult_iff: "coprime (k::int) n \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   448
    (k dvd m * n) = (k dvd m)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   449
  by (auto intro: int_coprime_dvd_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   450
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   451
lemma nat_gcd_mult_cancel: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   452
  apply (rule dvd_anti_sym)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   453
  apply (rule nat_gcd_greatest)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   454
  apply (rule_tac n = k in nat_coprime_dvd_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   455
  apply (simp add: nat_gcd_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   456
  apply (simp add: nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   457
  apply (simp_all add: mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   458
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   459
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   460
lemma int_gcd_mult_cancel:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   461
  assumes "coprime (k::int) n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   462
  shows "gcd (k * m) n = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   463
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   464
  using prems
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   465
  apply (subst (1 2) int_gcd_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   466
  apply (subst abs_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   467
  apply (rule nat_gcd_mult_cancel [transferred])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   468
  apply (auto simp add: int_gcd_abs [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   469
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   470
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   471
text {* \medskip Addition laws *}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   472
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   473
lemma nat_gcd_add1 [simp]: "gcd ((m::nat) + n) n = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   474
  apply (case_tac "n = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   475
  apply (simp_all add: nat_gcd_non_0)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   476
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   477
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   478
lemma nat_gcd_add2 [simp]: "gcd (m::nat) (m + n) = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   479
  apply (subst (1 2) nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   480
  apply (subst add_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   481
  apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   482
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   483
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   484
(* to do: add the other variations? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   485
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   486
lemma nat_gcd_diff1: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   487
  by (subst nat_gcd_add1 [symmetric], auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   488
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   489
lemma nat_gcd_diff2: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   490
  apply (subst nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   491
  apply (subst nat_gcd_diff1 [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   492
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   493
  apply (subst nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   494
  apply (subst nat_gcd_diff1)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   495
  apply assumption
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   496
  apply (rule nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   497
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   498
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   499
lemma int_gcd_non_0: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   500
  apply (frule_tac b = y and a = x in pos_mod_sign)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   501
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   502
  apply (auto simp add: nat_gcd_non_0 nat_mod_distrib [symmetric]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   503
    zmod_zminus1_eq_if)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   504
  apply (frule_tac a = x in pos_mod_bound)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   505
  apply (subst (1 2) nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   506
  apply (simp del: pos_mod_bound add: nat_diff_distrib nat_gcd_diff2
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   507
    nat_le_eq_zle)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   508
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   509
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   510
lemma int_gcd_red: "gcd (x::int) y = gcd y (x mod y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   511
  apply (case_tac "y = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   512
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   513
  apply (case_tac "y > 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   514
  apply (subst int_gcd_non_0, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   515
  apply (insert int_gcd_non_0 [of "-y" "-x"])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   516
  apply (auto simp add: int_gcd_neg1 int_gcd_neg2)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   517
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   518
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   519
lemma int_gcd_add1 [simp]: "gcd ((m::int) + n) n = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   520
  apply (case_tac "n = 0", force)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   521
  apply (subst (1 2) int_gcd_red)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   522
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   523
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   524
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   525
lemma int_gcd_add2 [simp]: "gcd m ((m::int) + n) = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   526
  apply (subst int_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   527
  apply (subst add_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   528
  apply (subst int_gcd_add1)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   529
  apply (subst int_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   530
  apply (rule refl)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   531
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   532
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   533
lemma nat_gcd_add_mult: "gcd (m::nat) (k * m + n) = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   534
  by (induct k, simp_all add: ring_simps)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   535
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   536
lemma int_gcd_add_mult: "gcd (m::int) (k * m + n) = gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   537
  apply (subgoal_tac "ALL s. ALL m. ALL n.
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   538
      gcd m (int (s::nat) * m + n) = gcd m n")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   539
  apply (case_tac "k >= 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   540
  apply (drule_tac x = "nat k" in spec, force)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   541
  apply (subst (1 2) int_gcd_neg2 [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   542
  apply (drule_tac x = "nat (- k)" in spec)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   543
  apply (drule_tac x = "m" in spec)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   544
  apply (drule_tac x = "-n" in spec)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   545
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   546
  apply (rule nat_induct)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   547
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   548
  apply (auto simp add: left_distrib)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   549
  apply (subst add_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   550
  apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   551
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   552
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   553
(* to do: differences, and all variations of addition rules
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   554
    as simplification rules for nat and int *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   555
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   556
lemma nat_gcd_dvd_prod [iff]: "gcd (m::nat) n dvd k * n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   557
  using mult_dvd_mono [of 1] by auto
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   558
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   559
(* to do: add the three variations of these, and for ints? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   560
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   561
lemma finite_divisors_nat:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   562
  assumes "(m::nat)~= 0" shows "finite{d. d dvd m}"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   563
proof-
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   564
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   565
  from finite_subset[OF _ this] show ?thesis using assms
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   566
    by(bestsimp intro!:dvd_imp_le)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   567
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   568
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   569
lemma finite_divisors_int:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   570
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   571
proof-
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   572
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   573
  hence "finite{d. abs d <= abs i}" by simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   574
  from finite_subset[OF _ this] show ?thesis using assms
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   575
    by(bestsimp intro!:dvd_imp_le_int)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   576
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   577
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   578
lemma gcd_is_Max_divisors_nat:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   579
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   580
apply(rule Max_eqI[THEN sym])
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   581
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_nat)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   582
 apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   583
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le nat_gcd_greatest_iff nat_gcd_pos)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   584
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   585
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   586
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   587
lemma gcd_is_Max_divisors_int:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   588
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   589
apply(rule Max_eqI[THEN sym])
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   590
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_int)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   591
 apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   592
 apply (metis int_gcd_greatest_iff int_gcd_pos zdvd_imp_le)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   593
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   594
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   595
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   596
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   597
subsection {* Coprimality *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   598
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   599
lemma nat_div_gcd_coprime [intro]:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   600
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   601
  shows "coprime (a div gcd a b) (b div gcd a b)"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   602
proof -
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
   603
  let ?g = "gcd a b"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   604
  let ?a' = "a div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   605
  let ?b' = "b div ?g"
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
   606
  let ?g' = "gcd ?a' ?b'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   607
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   608
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   609
  from dvdg dvdg' obtain ka kb ka' kb' where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   610
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   611
    unfolding dvd_def by blast
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   612
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   613
    by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   614
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   615
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   616
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   617
  have "?g \<noteq> 0" using nz by (simp add: nat_gcd_zero)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   618
  then have gp: "?g > 0" by arith
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   619
  from nat_gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   620
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   621
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   622
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   623
lemma int_div_gcd_coprime [intro]:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   624
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   625
  shows "coprime (a div gcd a b) (b div gcd a b)"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   626
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   627
  apply (subst (1 2 3) int_gcd_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   628
  apply (subst (1 2) abs_div)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   629
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   630
  prefer 3
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   631
  apply (rule nat_div_gcd_coprime [transferred])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   632
  using nz apply (auto simp add: int_gcd_abs [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   633
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   634
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   635
lemma nat_coprime: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   636
  using nat_gcd_unique[of 1 a b, simplified] by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   637
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   638
lemma nat_coprime_Suc_0:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   639
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   640
  using nat_coprime by (simp add: One_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   641
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   642
lemma int_coprime: "coprime (a::int) b \<longleftrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   643
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   644
  using int_gcd_unique [of 1 a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   645
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   646
  apply (erule subst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   647
  apply (rule iffI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   648
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   649
  apply (drule_tac x = "abs e" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   650
  apply (case_tac "e >= 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   651
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   652
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   653
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   654
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   655
lemma nat_gcd_coprime:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   656
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   657
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   658
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   659
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   660
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   661
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   662
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   663
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   664
  apply (rule nat_div_gcd_coprime)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   665
  using prems
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   666
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   667
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   668
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   669
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   670
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   671
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   672
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   673
lemma int_gcd_coprime:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   674
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   675
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   676
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   677
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   678
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   679
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   680
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   681
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   682
  apply (rule int_div_gcd_coprime)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   683
  using prems
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   684
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   685
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   686
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   687
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   688
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   689
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   690
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   691
lemma nat_coprime_mult: assumes da: "coprime (d::nat) a" and db: "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   692
    shows "coprime d (a * b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   693
  apply (subst nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   694
  using da apply (subst nat_gcd_mult_cancel)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   695
  apply (subst nat_gcd_commute, assumption)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   696
  apply (subst nat_gcd_commute, rule db)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   697
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   698
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   699
lemma int_coprime_mult: assumes da: "coprime (d::int) a" and db: "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   700
    shows "coprime d (a * b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   701
  apply (subst int_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   702
  using da apply (subst int_gcd_mult_cancel)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   703
  apply (subst int_gcd_commute, assumption)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   704
  apply (subst int_gcd_commute, rule db)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   705
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   707
lemma nat_coprime_lmult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   708
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   709
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   710
  have "gcd d a dvd gcd d (a * b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   711
    by (rule nat_gcd_greatest, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   712
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   713
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   714
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   715
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   716
lemma int_coprime_lmult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   717
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d a"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   718
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   719
  have "gcd d a dvd gcd d (a * b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   720
    by (rule int_gcd_greatest, auto)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   721
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   722
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   723
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   724
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   725
lemma nat_coprime_rmult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   726
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   727
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   728
  have "gcd d b dvd gcd d (a * b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   729
    by (rule nat_gcd_greatest, auto intro: dvd_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   730
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   731
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   732
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   733
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   734
lemma int_coprime_rmult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   735
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   736
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   737
  have "gcd d b dvd gcd d (a * b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   738
    by (rule int_gcd_greatest, auto intro: dvd_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   739
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   740
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   741
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   742
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   743
lemma nat_coprime_mul_eq: "coprime (d::nat) (a * b) \<longleftrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   744
    coprime d a \<and>  coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   745
  using nat_coprime_rmult[of d a b] nat_coprime_lmult[of d a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   746
    nat_coprime_mult[of d a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   747
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   748
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   749
lemma int_coprime_mul_eq: "coprime (d::int) (a * b) \<longleftrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   750
    coprime d a \<and>  coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   751
  using int_coprime_rmult[of d a b] int_coprime_lmult[of d a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   752
    int_coprime_mult[of d a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   753
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   754
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   755
lemma nat_gcd_coprime_exists:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   756
    assumes nz: "gcd (a::nat) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   757
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   758
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   759
  apply (rule_tac x = "b div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   760
  using nz apply (auto simp add: nat_div_gcd_coprime dvd_div_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   761
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   762
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   763
lemma int_gcd_coprime_exists:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   764
    assumes nz: "gcd (a::int) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   765
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   766
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   767
  apply (rule_tac x = "b div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   768
  using nz apply (auto simp add: int_div_gcd_coprime dvd_div_mult_self)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   769
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   770
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   771
lemma nat_coprime_exp: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   772
  by (induct n, simp_all add: nat_coprime_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   773
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   774
lemma int_coprime_exp: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   775
  by (induct n, simp_all add: int_coprime_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   776
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   777
lemma nat_coprime_exp2 [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   778
  apply (rule nat_coprime_exp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   779
  apply (subst nat_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   780
  apply (rule nat_coprime_exp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   781
  apply (subst nat_gcd_commute, assumption)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   782
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   783
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   784
lemma int_coprime_exp2 [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   785
  apply (rule int_coprime_exp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   786
  apply (subst int_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   787
  apply (rule int_coprime_exp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   788
  apply (subst int_gcd_commute, assumption)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   789
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   790
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   791
lemma nat_gcd_exp: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   792
proof (cases)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   793
  assume "a = 0 & b = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   794
  thus ?thesis by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   795
  next assume "~(a = 0 & b = 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   796
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   797
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   798
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   799
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   800
    apply (subst (1 2) mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   801
    apply (subst nat_gcd_mult_distrib [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   802
    apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   803
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   804
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   805
    apply (subst div_power)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   806
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   807
    apply (rule dvd_div_mult_self)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   808
    apply (rule dvd_power_same)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   809
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   810
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   811
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   812
    apply (subst div_power)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   813
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   814
    apply (rule dvd_div_mult_self)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   815
    apply (rule dvd_power_same)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   816
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   817
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   818
  finally show ?thesis .
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   819
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   820
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   821
lemma int_gcd_exp: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   822
  apply (subst (1 2) int_gcd_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   823
  apply (subst (1 2) power_abs)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   824
  apply (rule nat_gcd_exp [where n = n, transferred])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   825
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   826
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   827
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   828
lemma nat_coprime_divprod: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   829
  using nat_coprime_dvd_mult_iff[of d a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   830
  by (auto simp add: mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   831
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   832
lemma int_coprime_divprod: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   833
  using int_coprime_dvd_mult_iff[of d a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   834
  by (auto simp add: mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   835
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   836
lemma nat_division_decomp: assumes dc: "(a::nat) dvd b * c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   837
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   838
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   839
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   840
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   841
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   842
  {assume z: "?g \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   843
    from nat_gcd_coprime_exists[OF z]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   844
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   845
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   846
    have thb: "?g dvd b" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   847
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   848
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   849
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   850
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   851
    with z have th_1: "a' dvd b' * c" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   852
    from nat_coprime_dvd_mult[OF ab'(3)] th_1
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   853
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   854
    from ab' have "a = ?g*a'" by algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   855
    with thb thc have ?thesis by blast }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   856
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   857
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   858
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   859
lemma int_division_decomp: assumes dc: "(a::int) dvd b * c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   860
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   861
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   862
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   863
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   864
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   865
  {assume z: "?g \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   866
    from int_gcd_coprime_exists[OF z]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   867
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   868
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   869
    have thb: "?g dvd b" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   870
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   871
    with dc have th0: "a' dvd b*c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   872
      using dvd_trans[of a' a "b*c"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   873
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   874
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   875
    with z have th_1: "a' dvd b' * c" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   876
    from int_coprime_dvd_mult[OF ab'(3)] th_1
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   877
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   878
    from ab' have "a = ?g*a'" by algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   879
    with thb thc have ?thesis by blast }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   880
  ultimately show ?thesis by blast
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   881
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   882
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   883
lemma nat_pow_divides_pow:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   884
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   885
  shows "a dvd b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   886
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   887
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   888
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   889
  {assume "?g = 0" with ab n have ?thesis by auto }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   890
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   891
  {assume z: "?g \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   892
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   893
    from nat_gcd_coprime_exists[OF z]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   894
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   895
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   896
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   897
      by (simp add: ab'(1,2)[symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   898
    hence "?g^n*a'^n dvd ?g^n *b'^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   899
      by (simp only: power_mult_distrib mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   900
    with zn z n have th0:"a'^n dvd b'^n" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   901
    have "a' dvd a'^n" by (simp add: m)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   902
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   903
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   904
    from nat_coprime_dvd_mult[OF nat_coprime_exp [OF ab'(3), of m]] th1
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   905
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   906
    hence "a'*?g dvd b'*?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   907
    with ab'(1,2)  have ?thesis by simp }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   908
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   909
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   910
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   911
lemma int_pow_divides_pow:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   912
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   913
  shows "a dvd b"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   914
proof-
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   915
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   916
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   917
  {assume "?g = 0" with ab n have ?thesis by auto }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   918
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   919
  {assume z: "?g \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   920
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   921
    from int_gcd_coprime_exists[OF z]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   922
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   923
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   924
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   925
      by (simp add: ab'(1,2)[symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   926
    hence "?g^n*a'^n dvd ?g^n *b'^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   927
      by (simp only: power_mult_distrib mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   928
    with zn z n have th0:"a'^n dvd b'^n" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   929
    have "a' dvd a'^n" by (simp add: m)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   930
    with th0 have "a' dvd b'^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   931
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   932
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   933
    from int_coprime_dvd_mult[OF int_coprime_exp [OF ab'(3), of m]] th1
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   934
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   935
    hence "a'*?g dvd b'*?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   936
    with ab'(1,2)  have ?thesis by simp }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   937
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   938
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   939
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   940
lemma nat_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   941
  by (auto intro: nat_pow_divides_pow dvd_power_same)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   942
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   943
lemma int_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   944
  by (auto intro: int_pow_divides_pow dvd_power_same)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   945
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   946
lemma nat_divides_mult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   947
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   948
  shows "m * n dvd r"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   949
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   950
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   951
    unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   952
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   953
  hence "m dvd n'" using nat_coprime_dvd_mult_iff[OF mn] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   954
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   955
  from n' k show ?thesis unfolding dvd_def by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   956
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   957
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   958
lemma int_divides_mult:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   959
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   960
  shows "m * n dvd r"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   961
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   962
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   963
    unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   964
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   965
  hence "m dvd n'" using int_coprime_dvd_mult_iff[OF mn] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   966
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   967
  from n' k show ?thesis unfolding dvd_def by auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   968
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   969
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   970
lemma nat_coprime_plus_one [simp]: "coprime ((n::nat) + 1) n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   971
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   972
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   973
  apply (rule nat_dvd_diff)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   974
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   975
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   976
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   977
lemma nat_coprime_Suc [simp]: "coprime (Suc n) n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   978
  using nat_coprime_plus_one by (simp add: One_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   979
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   980
lemma int_coprime_plus_one [simp]: "coprime ((n::int) + 1) n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   981
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   982
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   983
  apply (rule dvd_diff)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   984
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   985
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   986
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   987
lemma nat_coprime_minus_one: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   988
  using nat_coprime_plus_one [of "n - 1"]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   989
    nat_gcd_commute [of "n - 1" n] by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   990
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   991
lemma int_coprime_minus_one: "coprime ((n::int) - 1) n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   992
  using int_coprime_plus_one [of "n - 1"]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   993
    int_gcd_commute [of "n - 1" n] by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   994
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   995
lemma nat_setprod_coprime [rule_format]:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   996
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   997
  apply (case_tac "finite A")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   998
  apply (induct set: finite)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   999
  apply (auto simp add: nat_gcd_mult_cancel)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1000
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1001
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1002
lemma int_setprod_coprime [rule_format]:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1003
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1004
  apply (case_tac "finite A")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1005
  apply (induct set: finite)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1006
  apply (auto simp add: int_gcd_mult_cancel)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1007
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1008
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1009
lemma nat_prime_odd: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1010
  unfolding prime_nat_def
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1011
  apply (subst even_mult_two_ex)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1012
  apply clarify
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1013
  apply (drule_tac x = 2 in spec)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1014
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1015
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1016
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1017
lemma int_prime_odd: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1018
  unfolding prime_int_def
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1019
  apply (frule nat_prime_odd)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1020
  apply (auto simp add: even_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1021
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1022
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1023
lemma nat_coprime_common_divisor: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1024
    x dvd b \<Longrightarrow> x = 1"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1025
  apply (subgoal_tac "x dvd gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1026
  apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1027
  apply (erule (1) nat_gcd_greatest)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1028
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1029
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1030
lemma int_coprime_common_divisor: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1031
    x dvd b \<Longrightarrow> abs x = 1"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1032
  apply (subgoal_tac "x dvd gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1033
  apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1034
  apply (erule (1) int_gcd_greatest)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1035
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1036
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1037
lemma nat_coprime_divisors: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1038
    coprime d e"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1039
  apply (auto simp add: dvd_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1040
  apply (frule int_coprime_lmult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1041
  apply (subst int_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1042
  apply (subst (asm) (2) int_gcd_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1043
  apply (erule int_coprime_lmult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1044
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1045
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1046
lemma nat_invertible_coprime: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1047
apply (metis nat_coprime_lmult nat_gcd_1 nat_gcd_commute nat_gcd_red)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1048
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1049
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1050
lemma int_invertible_coprime: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1051
apply (metis int_coprime_lmult int_gcd_1 int_gcd_commute int_gcd_red)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1052
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1053
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1054
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1055
subsection {* Bezout's theorem *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1056
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1057
(* Function bezw returns a pair of witnesses to Bezout's theorem --
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1058
   see the theorems that follow the definition. *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1059
fun
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1060
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1061
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1062
  "bezw x y =
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1063
  (if y = 0 then (1, 0) else
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1064
      (snd (bezw y (x mod y)),
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1065
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1066
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1067
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1068
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1069
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1070
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1071
  by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1072
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1073
declare bezw.simps [simp del]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1074
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1075
lemma bezw_aux [rule_format]:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1076
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1077
proof (induct x y rule: nat_gcd_induct)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1078
  fix m :: nat
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1079
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1080
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1081
  next fix m :: nat and n
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1082
    assume ngt0: "n > 0" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1083
      ih: "fst (bezw n (m mod n)) * int n +
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1084
        snd (bezw n (m mod n)) * int (m mod n) =
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1085
        int (gcd n (m mod n))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1086
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1087
      apply (simp add: bezw_non_0 nat_gcd_non_0)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1088
      apply (erule subst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1089
      apply (simp add: ring_simps)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1090
      apply (subst mod_div_equality [of m n, symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1091
      (* applying simp here undoes the last substitution!
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1092
         what is procedure cancel_div_mod? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1093
      apply (simp only: ring_simps zadd_int [symmetric]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1094
        zmult_int [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1095
      done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1096
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1097
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1098
lemma int_bezout:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1099
  fixes x y
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1100
  shows "EX u v. u * (x::int) + v * y = gcd x y"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1101
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1102
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1103
      EX u v. u * x + v * y = gcd x y"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1104
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1105
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1106
    apply (unfold gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1107
    apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1108
    apply (subst bezw_aux [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1109
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1110
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1111
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1112
      (x \<le> 0 \<and> y \<le> 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1113
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1114
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1115
    by (erule (1) bezout_aux)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1116
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1117
    apply (insert bezout_aux [of x "-y"])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1118
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1119
    apply (rule_tac x = u in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1120
    apply (rule_tac x = "-v" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1121
    apply (subst int_gcd_neg2 [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1122
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1123
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1124
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1125
    apply (insert bezout_aux [of "-x" y])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1126
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1127
    apply (rule_tac x = "-u" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1128
    apply (rule_tac x = v in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1129
    apply (subst int_gcd_neg1 [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1130
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1131
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1132
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1133
    apply (insert bezout_aux [of "-x" "-y"])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1134
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1135
    apply (rule_tac x = "-u" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1136
    apply (rule_tac x = "-v" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1137
    apply (subst int_gcd_neg1 [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1138
    apply (subst int_gcd_neg2 [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1139
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1140
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1141
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1142
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1143
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1144
text {* versions of Bezout for nat, by Amine Chaieb *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1145
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1146
lemma ind_euclid:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1147
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1148
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1149
  shows "P a b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1150
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1151
  fix n a b
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1152
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1153
  have "a = b \<or> a < b \<or> b < a" by arith
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1154
  moreover {assume eq: "a= b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1155
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1156
    by simp}
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1157
  moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1158
  {assume lt: "a < b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1159
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1160
    moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1161
    {assume "a =0" with z c have "P a b" by blast }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1162
    moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1163
    {assume ab: "a + b - a < n"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1164
      have th0: "a + b - a = a + (b - a)" using lt by arith
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1165
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1166
      have "P a b" by (simp add: th0[symmetric])}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1167
    ultimately have "P a b" by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1168
  moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1169
  {assume lt: "a > b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1170
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1171
    moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1172
    {assume "b =0" with z c have "P a b" by blast }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1173
    moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1174
    {assume ab: "b + a - b < n"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1175
      have th0: "b + a - b = b + (a - b)" using lt by arith
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1176
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1177
      have "P b a" by (simp add: th0[symmetric])
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1178
      hence "P a b" using c by blast }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1179
    ultimately have "P a b" by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1180
ultimately  show "P a b" by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1181
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1182
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1183
lemma nat_bezout_lemma:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1184
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1185
    (a * x = b * y + d \<or> b * x = a * y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1186
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1187
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1188
  using ex
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1189
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1190
  apply (rule_tac x="d" in exI, simp add: dvd_add)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1191
  apply (case_tac "a * x = b * y + d" , simp_all)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1192
  apply (rule_tac x="x + y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1193
  apply (rule_tac x="y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1194
  apply algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1195
  apply (rule_tac x="x" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1196
  apply (rule_tac x="x + y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1197
  apply algebra
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1198
done
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1199
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1200
lemma nat_bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1201
    (a * x = b * y + d \<or> b * x = a * y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1202
  apply(induct a b rule: ind_euclid)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1203
  apply blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1204
  apply clarify
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1205
  apply (rule_tac x="a" in exI, simp add: dvd_add)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1206
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1207
  apply (rule_tac x="d" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1208
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1209
  apply (rule_tac x="x+y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1210
  apply (rule_tac x="y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1211
  apply algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1212
  apply (rule_tac x="x" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1213
  apply (rule_tac x="x+y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1214
  apply algebra
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1215
done
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1216
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1217
lemma nat_bezout1: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1218
    (a * x - b * y = d \<or> b * x - a * y = d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1219
  using nat_bezout_add[of a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1220
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1221
  apply (rule_tac x="d" in exI, simp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1222
  apply (rule_tac x="x" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1223
  apply (rule_tac x="y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1224
  apply auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1225
done
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1226
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1227
lemma nat_bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1228
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1229
proof-
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1230
 from nz have ap: "a > 0" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1231
 from nat_bezout_add[of a b]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1232
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1233
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1234
 moreover
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1235
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1236
     from H have ?thesis by blast }
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1237
 moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1238
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1239
   {assume b0: "b = 0" with H  have ?thesis by simp}
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1240
   moreover
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1241
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1242
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1243
       by auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1244
     moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1245
     {assume db: "d=b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1246
       from prems have ?thesis apply simp
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1247
	 apply (rule exI[where x = b], simp)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1248
	 apply (rule exI[where x = b])
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1249
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1250
    moreover
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1251
    {assume db: "d < b"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1252
	{assume "x=0" hence ?thesis  using prems by simp }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1253
	moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1254
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1255
	  from db have "d \<le> b - 1" by simp
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1256
	  hence "d*b \<le> b*(b - 1)" by simp
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1257
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1258
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1259
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1260
            by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1261
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1262
	    by (simp only: mult_assoc right_distrib)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1263
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1264
            by algebra
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1265
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1266
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1267
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1268
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1269
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1270
	  hence ?thesis using H(1,2)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1271
	    apply -
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1272
	    apply (rule exI[where x=d], simp)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1273
	    apply (rule exI[where x="(b - 1) * y"])
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1274
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1275
	ultimately have ?thesis by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1276
    ultimately have ?thesis by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1277
  ultimately have ?thesis by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1278
 ultimately show ?thesis by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1279
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1280
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1281
lemma nat_bezout: assumes a: "(a::nat) \<noteq> 0"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1282
  shows "\<exists>x y. a * x = b * y + gcd a b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1283
proof-
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1284
  let ?g = "gcd a b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1285
  from nat_bezout_add_strong[OF a, of b]
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1286
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1287
  from d(1,2) have "d dvd ?g" by simp
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1288
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1289
  from d(3) have "a * x * k = (b * y + d) *k " by auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1290
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1291
  thus ?thesis by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1292
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1293
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1294
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1295
subsection {* LCM *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1296
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1297
lemma int_lcm_altdef: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1298
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1299
    zmult_int [symmetric] gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1300
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1301
lemma nat_prod_gcd_lcm: "(m::nat) * n = gcd m n * lcm m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1302
  unfolding lcm_nat_def
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1303
  by (simp add: dvd_mult_div_cancel [OF nat_gcd_dvd_prod])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1304
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1305
lemma int_prod_gcd_lcm: "abs(m::int) * abs n = gcd m n * lcm m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1306
  unfolding lcm_int_def gcd_int_def
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1307
  apply (subst int_mult [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1308
  apply (subst nat_prod_gcd_lcm [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1309
  apply (subst nat_abs_mult_distrib [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1310
  apply (simp, simp add: abs_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1311
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1312
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1313
lemma nat_lcm_0 [simp]: "lcm (m::nat) 0 = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1314
  unfolding lcm_nat_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1315
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1316
lemma int_lcm_0 [simp]: "lcm (m::int) 0 = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1317
  unfolding lcm_int_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1318
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1319
lemma nat_lcm_1 [simp]: "lcm (m::nat) 1 = m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1320
  unfolding lcm_nat_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1321
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1322
lemma nat_lcm_Suc_0 [simp]: "lcm (m::nat) (Suc 0) = m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1323
  unfolding lcm_nat_def by (simp add: One_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1324
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1325
lemma int_lcm_1 [simp]: "lcm (m::int) 1 = abs m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1326
  unfolding lcm_int_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1327
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1328
lemma nat_lcm_0_left [simp]: "lcm (0::nat) n = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1329
  unfolding lcm_nat_def by simp
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1330
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1331
lemma int_lcm_0_left [simp]: "lcm (0::int) n = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1332
  unfolding lcm_int_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1333
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1334
lemma nat_lcm_1_left [simp]: "lcm (1::nat) m = m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1335
  unfolding lcm_nat_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1336
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1337
lemma nat_lcm_Suc_0_left [simp]: "lcm (Suc 0) m = m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1338
  unfolding lcm_nat_def by (simp add: One_nat_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1339
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1340
lemma int_lcm_1_left [simp]: "lcm (1::int) m = abs m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1341
  unfolding lcm_int_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1342
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1343
lemma nat_lcm_commute: "lcm (m::nat) n = lcm n m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1344
  unfolding lcm_nat_def by (simp add: nat_gcd_commute ring_simps)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1345
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1346
lemma int_lcm_commute: "lcm (m::int) n = lcm n m"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1347
  unfolding lcm_int_def by (subst nat_lcm_commute, rule refl)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1348
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1349
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1350
lemma nat_lcm_pos:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1351
  assumes mpos: "(m::nat) > 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1352
  and npos: "n>0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1353
  shows "lcm m n > 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1354
proof(rule ccontr, simp add: lcm_nat_def nat_gcd_zero)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1355
  assume h:"m*n div gcd m n = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1356
  from mpos npos have "gcd m n \<noteq> 0" using nat_gcd_zero by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1357
  hence gcdp: "gcd m n > 0" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1358
  with h
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1359
  have "m*n < gcd m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1360
    by (cases "m * n < gcd m n")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1361
       (auto simp add: div_if[OF gcdp, where m="m*n"])
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1362
  moreover
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1363
  have "gcd m n dvd m" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1364
  with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1365
  with npos have t1:"gcd m n*n \<le> m*n" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset